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Quantum Spacetime, Quantum Geometry and Planck scales
Journal Pre-proof Quantum Spacetime, Quantum Geometry and Planck scales Sergio Doplicher
PII: DOI: Reference:
S0723-0869(20)30002-5 https://doi.org/10.1016/j.exmath.2020.01.002 EXMATH 25383
To appear in:
Expositiones Mathematicae
Please cite this article as: S. Doplicher, Quantum Spacetime, Quantum Geometry and Planck scales, Expositiones Mathematicae (2020), doi: https://doi.org/10.1016/j.exmath.2020.01.002. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. c 2020 Published by Elsevier GmbH. ⃝
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In memoriam Dick Kadison
Sergio Doplicher
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Quantum Spacetime, Quantum Geometry and Planck scales
Dipartimento di Matematica, Universit` a di Roma “La Sapienza”, Piazzale Aldo Moro, 5, I-00185 Roma, Italy, email [email protected].
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Introduction
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Noncommutativity is the main feature of Operator Algebras, a domain where Dick Kadison has been a giant. For these reason maybe it is not unappropriate to honor his memory talking of a new aspect of noncommutativity, which emerged in Physics in relatively recent times; after some heuristic motivation in this introduction, we outline some of its mathematical and physical implications in the next sections. It is well known that noncommutativity is the central feature of Quantum Mechanics; for systems with finitely many degrees of freedom, the Heisenberg uncertainty principle ∆q∆p & ~ relating the uncertainty in the measurement of a coordinate q of the position of a particle to that of its conjugate momentum p, are a basic physical principle, which is mathematically implemented by the Born - Heisenberg commutation relations pq − qp = −i~.
In Quantum Field Theory the basic observable quantities are the local observables: those which can be measured within the spacetime specifications given by a bounded open region (say a double cone) O in spacetime, generate a von Neumann algebra A(O), acting on the Hilbert space of the vacuum superselection sector, where the C* algebra generated for all O acts irreducibly. The basic principle of Locality requires that
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AB = BA
whenever
A ∈ A(O1 ), B ∈ A(O2 , ),
and O1 , O2 are spacelike separated, i.e. they cannot be connected by signals traveling not faster than light. This principle is consistent with what can be observed at all accessible scales; in Quantum Field Theory it holds at all scales, if we neglect the gravitational forces
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between elementary particles (which, of course, at today accessible distances, are weak beyond any imaginable level). If we do not, we must remember that, according to the Heisenberg uncertainty principle, localizing an event in a small region costs energy, and that, according to Einstein, energy generates a gravitational field. Thus the concurrence of Quantum Mechanics and of General Relativity leads us to the P rinciple of Gravitational Stability against localization of events [6]: The gravitational field generated by the concentration of energy required by the Heisenberg Uncertainty Principle to localize an event in spacetime should not be so strong to hide the event itself to any distant observer - distant compared to the Planck scale. To indicate how this principle limits the precision in localizing an event, recall that a spherically symmetric localization in space with accuracy a implies that an uncontrollable energy E of order 1/a has to be transferred (we use universal units where the Plank constant ~, the speed of light c, and the gravitational constant G are equal to 1). The associated Schwarzschild radius R, giving the size of a sphere around the origin out of which no signal can emerge, is R'E+U
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if U is the energy already present at the observed spot, in a background spherically symmetric quantum state. Hence we must have that a & R ' 1/a + U ;
so that if U is much smaller than 1
a & 1,
the Planck Length; that means, in CGS units,
a & λP ' 1.6 · 10−33 cm;
if U is much larger than 1,
a & U,
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and the “minimal distance” is dynamical, the Effective Planck Length, which might diverge. Which shows a possible path to solve the “Horizon Problem” (why the Cosmic Microwave Background has the same temperature in all directions, showing the reach of thermodynamical equilibrium even for regions which were never connected by non - superluminal signals?): divergent Effective Planck Length means instant long range (a-causal) correlations, allowing establishment of thermal equilibrium [3]. Such an explanation would not need the hypothesis of inflation.
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But at t = 0 all points are instantly connected to one another: a single point. The number of degrees of freedom is collapsing to zero? This point is discussed in [1]. Anyway, at least at distances of the order of the Planck length λP , causality, hence Locality, is expected to fail. The paradise, where an impressive part of the mathematical and conceptual structure of Quantum Field Theory derives essentially from Locality alone [11], is thus lost. One might hope that, at the Planck scale, a yet unknown but equally sharp Physical Principle replaces Locality, reducing to it at distances which are large compared with the Planck Length. We comment on this point at the end of this note. Let us now focus on the case where the local energy content U of the background state can be neglected, but we do not limit ourselves to the case of spherical symmetry. If we measure one or at most two space coordinates with great precision a, but the uncertainty L in another coordinates is large, the energy 1/a may spread over a region of size L, and generate a gravitational potential that vanishes everywhere as L → ∞ (provided a, as small as we like but non zero, remains constant). This indicates that the ∆q µ must satisfy Uncertainty Relations. Which should be then implemented by commutation relations. Giving rise to a Quantum Spacetime. But if we do not neglect the background energy U , there must be a dependence of the Uncertainty Relations, hence of the Commutators between coordinates, upon the background quantum state i.e. upon the metric tensor. Thus, while Classical General Relativity says that Geometry is Dynamics, a Quantum theory of Gravity is to be expected to say that also the Algebra of Spacetime is Dynamics.
Quantum Minkowski Space
The Principle of Gravitational Stability against localization of events implies the following Spacetime Uncertainty Relations (STUR):
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∆q0 ·
3 X j=1
X
∆qj & 1;
∆qj ∆qk & 1.
(1)
1≤j
first deduced using the linear approximation of the Einstein Equations [6]; adopting instead the Hoop Conjecture, further evidence was provided [9, 10], extending them to a curved background as well. The notion of Energy, problematic in General Relativity, plays a role: but in the special case of spherically symmetric experiments, with all spacetime uncertainties
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taking all the same value, the exact semiclassical Einstein Equations, without any reference to energy, imply a minimal value of the proper length of order of the effective Planck length [3]. We come back to this point further below. Returning to Minkowski, the STUR must be implemented by spacetime commutation relations [qµ , qν ] = iλ2P Qµν
(2)
imposing Quantum Conditions on the Qµν . The simplest solution: [q µ , Qν,λ ] = 0; µν
Qµν Q
(3)
= 0;
(4)
2
((1/2) [q0 , . . . , q3 ]) = I,
(5)
where, under Lorentz transformations, Qµν Qµν is a scalar, and
q0 .. [q0 , . . . , q3 ] ≡ det . q0
··· .. . ···
q3 .. . q3
≡ εµνλρ qµ qν qλ qρ =
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= −(1/2)Qµν (∗Q)µν
(6)
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is a pseudoscalar, hence we use the square in the Quantum Conditions. The so obtained “basic model of Quantum Spacetime” implements exactly the Spacetime Uncertainty Relations and is fully Poincar´e covariant. The classical Poincar´e group acts as symmetries; translations, in particular, act adding to each qµ a real multiple of the identity. The noncommutative C* algebra of Quantum Spacetime can be associated to the above relations. The procedure [6], which applies to more general cases, in short goes as follows. Assuming that the qλ , Qµν are selfadjoint operators and that the Qµν commute strongly with one another and with the qλ , the relations above can be seen as a bundle of Lie Algebra relations based on the joint spectrum Σ of the Qµν . Irreducible representations must be concentrated at points of Σ, providing representations of those Lie algebras; the regular representations must be integrable, hence described by representations of the C* group algebra of the unique simply connected Lie group associated to the corresponding Lie algebra. Thus the C* algebra of Quantum Spacetime is the C* algebra of a continuous field of group C* algebras based on the spectrum of a commutative C* algebra. In our case, that spectrum - the joint spectrum of the Qµν - is the manifold Σ of the real valued antisymmetric 2 - tensors fulfilling the same relations as the Qµν do: a homogeneous space of the proper orthochronous Lorentz group, identified with the coset space of SL(2, C) mod the subgroup of diagonal matrices. Each of
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those tensors, can be taken to its rest frame, where the “electric ”and “magnetic ”parts e, m are parallel unit vectors, by a boost. The inverse boost is specified by a third vector, orthogonal to those unit vectors; thus Σ can be viewed as the tangent bundle to two copies of the unit sphere in 3 space - its base Σ1 [5]. The irreducible representations at a point of Σ1 provide the Shroedinger operators p, q in 2 degrees of freedom. The fibers, with the condition that I is not an independent generator but is represented by I, coincide with the C* algebra of the Born - Heisenberg relations in 2 degrees of freedom - the algebra of all compact operators on a fixed infinite dimensional separable Hilbert space. The continuous field can be shown to be trivial. Thus the C* algebra E of Quantum Spacetime is identified with the tensor product of the continuous functions vanishing at infinity on Σ and the algebra of compact operators [6]. The mathematical generalization of points are pure states. The Optimally localized states ω are those minimizing the quantity Σµ (∆ω qµ )2 ;
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the minimum is 2, and is actually reached by states concentrated on Σ1 , at each point coinciding with the ground state of the harmonic oscillator . More precisely, these states are given by an optimal localization map composed with a probability measure on Σ1 . But to explore more thoroughly the Quantum Geometry of Quantum Spacetime we must consider independent events. Quantum mechanically n independent events ought to be described by the n − f old tensor product of E with itself; considering arbitrary values of n we are led to use the direct sum over all n. If A is the C* algebra with unit over C, obtained adding the unit to E, we will view the n-fold tensor power Λn (A) of A over C as an A-bimodule with the product in A, a(a1 ⊗ a2 ⊗ ... ⊗ an ) = (aa1 ) ⊗ a2 ⊗ ... ⊗ an ;
(a1 ⊗ a2 ⊗ ... ⊗ an )a = a1 ⊗ a2 ⊗ ... ⊗ (an a);
and the direct sum
Λ(A) =
∞ M
Λn (A)
n=0
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as the A-bimodule tensor algebra,
(a1 ⊗ a2 ⊗ ... ⊗ an )(b1 ⊗ b2 ⊗ ... ⊗ bm ) = a1 ⊗ a2 ⊗ ... ⊗ (an b1 ) ⊗ b2 ⊗ ... ⊗ bm .
This is the natural ambient for the universal differential calculus, where the differential is given by d(a0 ⊗ · · · ⊗ an ) =
n X
(−1)k a0 ⊗ · · · ⊗ ak−1 ⊗ I ⊗ ak ⊗ · · · ⊗ an .
k=0
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As usual d is a graded differential , i.e., if φ ∈ Λ(A), ψ ∈ Λn (A), we have
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d2 = 0;
d(φ · ψ) = (dφ) · ψ + (−1)n φ · dψ.
Note that A = Λ1 (A) ⊂ Λ(A), and the d-stable subalgebra Ω(A) of Λ(A) generated by A is the universal differential algebra with dI = 0. In other words, it is the subalgebra generated by A and da = I ⊗ a − a ⊗ I
as a varies in A. In the case of n independent events one is led to describe the spacetime coordinates of the j − th event by qj = I ⊗ ... ⊗ I ⊗ q ⊗ I... ⊗ I (q in the j - th place); in this way, the commutator between the different spacetime components of the qj would depend on j. A better choice is to require that it does not; this is achieved as follows. The centre Z of the multiplier algebra of E is the algebra of all bounded continuous complex functions on Σ; so that E, and hence A, is in an obvious way a Z − bimodule. We therefore can, and will, replace, in the definition of Λ(A), the C - tensor product, by the Z − bimodule − tensor product, so that
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dQ = 0.
As a consequence, the qj and the 2−1/2 (qj − qk ), j different from k, and the 2−1/2 dq, obey the same spacetime commutation relations, as does the normalized barycenter coordinates, n−1/2 (q1 + q2 + ... + qn ); and the latter commute with the difference coordinates. These facts allow us to define a quantum diagonal map from Λn (E) to E1 (the restriction to Σ1 of E), E (n) : E ⊗Z · · · ⊗Z E −→ E1
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which factors to that restriction map and a conditional expectation, which leaves the functions of the barycenter coordinates alone, and evaluates on functions of the difference variables the universal optimally localized map (which, when composed with a probability measure on Σ1 , would give the generic optimally localized state). Replacing the classical diagonal evaluation of a function of n arguments on Minkowski space by the quantum diagonal map allows us to define the Quantum W ick P roduct. But working in Ω(A) as a subspace of Λ(A) allows us to use two structures: - the tensor algebra structure described above, where both the A bimodule and the Z bimodule structures enter, essential for our reduced universal differential calculus; - the pre - C* algebra structure of Λ(A), which allows us to consider, for each element a of Λn (A), its modulus (a∗ a)1/2 , its spectrum, and so on.
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In particular we can study the geometric operators: separation between two independent events, area, 3 - volume, 4 - volume, given by dq;
dq ∧ dq;
dq ∧ dq ∧ dq;
dq ∧ dq ∧ dq ∧ dq, where, for instance, the latter is given by
V = dq ∧ dq ∧ dq ∧ dq = µνρσ dq µ dq ν dq ρ dq σ .
Each of these forms has a number of spacetime components: e.g. four for the first one (a vector), the last one is a pseudoscalar. Each component is shown to be a normal operator . Then we have the following Theorem [7]
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For each of these forms, the sum of the square moduli of all spacetime components is bounded below by a multiple of the identity of unit order of magnitude.
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Although that sum is (except for the 4 - volume!) not Lorentz invariant, the above mentioned bound holds in any Lorentz frame. In particular, - the four volume operator has pure point spectrum, at distance 51/2 − 2 from 0; - the Euclidean distance between two independent events has a lower bound of order one in Planck units. Thus, two distinct points can never merge to a point. However, of course, the state where the minimum is achieved will depend upon the reference frame where the requirement is formulated. (More details on the foregoing discussion may be found in [7] where the structure of length, area and volume operators on QST has been studied in full detail). Thus the existence of a minimal length is not at all in contradiction with the Lorentz covariance of the model. In the hope to use this formalism for Gauge Theory, a pairing was introduced on Λ(A) such that ha1 ⊗ a2 ⊗ .... ⊗ an , b1 ⊗ b2 ⊗ ... ⊗ bm i = δn,m T r(a1 b1 ...an bn )
where Tr is a faithful trace on A (here, any algebra with unit - and with a trace). Then, for any pair of differential forms φ, ψ on A,
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hφ, dψi = hδφ, ψi,
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where δ is the Hochschild boundary; which is then the Hodge dual of the exterior differential with respect to the above metric [7]. But unfortunately so far this formalism did not provide new insights on Gauge Theories on Quantum Spacetime. In the C* algebra E of Quantum Spacetime, we can define [6] the von Neumann functional calculus: for each f ∈ FL1 (R4 ) the function f (q) of the quantum coordinates qµ is given by Z µ f (q) ≡ fˇ(α)eiqµ α d4 α , and the integral over the whole space as well as that over 3 - space at q0 = t by
Z
Z
q0 =t
d4 qf (q) ≡ 3
f (q)d q ≡
Z
Z
f (x)d4 x = fˇ(0) = T rf (q), eik0 t fˇ(k0 , ~0)dk0 =
= lim T r(fm (q)∗ f (q)fm (q)), m
3
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where the trace is the ordinary trace at each point of the joint spectrum Σ of the commutators, i.e. a Z valued trace, and for suitable sequences of functions fm ∈ FL1 (R4 ). But on more general elements of our algebra both maps give Q - dependent results. These are important tools to define the interaction Hamiltonian to be used in the Gell’Mann Low formula for the S - Matrix.
Quantum Spacetime and Quantum Field Theory
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The geometry of Quantum Spacetime and the free field theories on it are fully Poincar´e covariant. One can introduce interactions in different ways, all preserving spacetime translation and space rotation covariance, all equivalent on ordinary Minkowski space, providing inequivalent approaches on QST; but all of them, sooner or later, meet problems with Lorentz covariance, apparently due to the nontrivial action of the Lorentz group on the centre of the multiplier algebra of Quantum Spacetime. On this points in our opinion a deeper understanding is needed. For instance, the interaction Hamiltonian on quantum spacetime Z HI (t) = λ d3 q : φ(q)n : q 0 =t
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would be Q - dependent; since there is no invariant probability measure or mean on Σ, we can do no better than integrating on Σ1 with the rotation invariant measure [6], and this breaks Lorentz invariance. Covariance is preserved by the Yang Feldmann equations, but missed again at the level of scattering theory. The Quantum Wick product selects a special frame from the start. The interaction Hamiltonian on the quantum spacetime is then given by Z HI (t) = λ d3 q : φ(q)n :Q q 0 =t
where
: φn (q) :Q = E (n) (: φ(q1 ) · · · φ(qn ) :)
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which does not depend on Q any longer, but brakes Lorentz invariance at an earlier stage. The last mentioned approach takes into account, in the very definition of Wick products, the fact that in our Quantum Spacetime n (larger or equal to two) distinct points can never merge to a point. But we can use the canonical quantum diagonal map which associates to functions of n independent points a function of a single point, evaluating a conditional expectation which on functions of the differences takes a numerical value, associated with the minimum of the Euclidean distance in a given Lorentz frame. The “Quantum Wick Product” obtained by this procedure leads to an interaction Hamiltonian on the quantum spacetime, given by the formula reported few line above, which is a constant operator–valued function of Σ1 (i.e. HI (t) is formally in the tensor product of C(Σ1 ) with the algebra of field operators). Thus one is lead to a unique prescription for the interaction Hamiltonian on quantum spacetime. When used in the Gell’Mann Low perturbative expansion for the S - Matrix, this gives the same result as the effective non local Hamiltonian determined by the kernel exp
n 1 X µ 2 (4) 1 X aj . a δ − 2 j,µ j n j=1
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The corresponding perturbative Gell’Mann Low formula is then free of ultraviolet divergences at each term of the perturbation expansion [8] . However, those terms have a meaning only after a sort of adiabatic cutoff: the coupling constant should be changed to a function of time, rapidly vanishing at infinity, say depending upon a cutoff time T. But the adiabatic limit T → ∞ is difficult problem. The use of the Quantum Wick product was dictated by the inconsistency with the quantum structure of spacetime of setting equal to zero the difference of coordinates of independent events. But there is another way to take care of the same problem: to replace, in the expression for the interaction Lagrangean density, the
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field operator φ(x) at a point x of the classical Minkowski space, with the evaluation of the field φ at a quantum point of QST, namely with < ι ⊗ ωx , φ(q) >
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where φ(q) is affiliated to F ⊗ E, F is the algebra of fields, ι is the identity map on F, and ωx the state of E optimally localized at x. Applied to a polynomial interaction, this procedure leads to another effective interaction Lagrangean on classical Minkowski space, whose integral over the whole spacetime agrees with that obtained using the Quantum Wick Product. But now the methods of Perturbative Algebraic Quantum Field Theory can be applied, and the perturbation expansion of the field operators is not only still ultraviolet finite, but also, in this case, converging term by term to an adiabatic limit when the time cutoff is removed [2]. This important fact allows us to consider the following problems. First, can one prove the convergence term by term to an adiabatic limit of the perturbation expansion of the S - matrix itself? If so, suppose we apply this construction to the renormalized Lagrangean of a theory which is renormalizable on the ordinary Minkowski space, with the counter terms defined by that ordinary theory, and with finite renormalization constants depending upon the Planck length λP . Those constants ought to be calculated for a generic value of λP by the requirement that the divergent terms of the key irreducible divergent Feynmann diagrams are cancelled. When λP → 0 , do we get back the ordinary renormalized expansion on Minkowski space? This would allow us to take the result of the described construction as a source of predictions to be compared with observations (cf [2, 1] for more details).
Quantum Spacetime and cosmology
The heuristic argument in the Introduction above suggest that the commutators between coordinates ought to depend on gµ,ν ; in that scenario the equations of motion sould include those commutation relations Rµ,ν − (1/2)Rgµ,ν = 8πTµ,ν (ψ);
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Fg (ψ) = 0;
µ
[q , q ν ] = iQµ,ν (g);
so that Algebra is Dynamics. And we can expect a dynamical minimal length. Which, in particular, would be divergent near singularities. This could solve Horizon Problem, without inflationary hypothesis. How solid are these heuristic arguments? We may consider as a model a massless scalar field with a semiclassical coupling with gravity; namely, we use the Quantum Wick product to define the Energy -
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Momentum Tensor TQµ,ν (q); we focus on the semiclassical Einstein Equations,with source ω ⊗ ηx (TQµ,ν (q)), where ω is a KMS state and ηx is a state on E optimally localized at x. If we assume full spherical symmetry the exact solution is available, and shows that the minimal proper length is at least λP [3]. Hence these simplifying ansaetze imply a solution describing spacetime without the horizon problem [3]. Near the Big Bang every pair of points were in causal contact, as indicated by the heuristic argument that the range of a-causal effects should diverge. For the Planck length λP is replaced by the effective Planck length λP /a(t), where a(t) is the coefficient in the FRW metric ds2 = −dt2 + a(t)2 (dx21 + dx22 + dx23 )
What happens at the singularity, t = 0? If we consider the Minkowski QST with varying effective Planck length, λeff → ∞,
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replacing λP by λeff → ∞ in the above formulas we get that, in that limit, the Quantum Diagonal Map vanishes: E (n) → 0. Interactions and the interacting fields themselves ought to vanish, a quantum point would be infinitely extended coinciding with the whole; a whole with zero degrees of freedom? This point and further consequences are discussed in [1]. But a fully convincing answer would require a dynamical picture of Quantum Spacetime, maybe in the scenario outlined at the beginning of this Section. Which might provide an answer to the question: are there observable signatures of QST? So far they seem too weak to be detected [4]. However, it has to be stressed that Quantum effects at Planck scale would depend upon an extrapolation of the Einstein Equations to that scale. But Newton’s law is experimentally checked only for distances not less than .01 centimeter! [12, 13], i.e. we are extrapolating 31 steps down in base 10 - logarithmic scale; while the size of the known universe is ”only” 28 steps up. Note that the observations with LHC in CERN on one side, and, on the other side, the nearly perfect agreement with Planck’s law of the spectral energy distribution of the Cosmic Microwave Background Radiation - the largest system in the universe - indicate the agreement with Quantum Mechanics of what can be observed in the range from 10−18 to 1028 centimeters. In the same range the Principle of Locality seems to hold unrestrictedly. But we saw it has to be expected that it will fail at the very short Planckian distances or near the big Bang; that is much below that lower limit, or about if not beyond that higher limit (for, due to its expansion, observing the known universe near its border would amount to observing it near the Big Bang). What would replace it? It might be helpful to wonder about the physical basis of that principle. Which seems to be the point - nature of field interactions, the great revolution of Maxwell’s viewpoint. That point - nature, in turn, is a consequence of the minimal form of interactions in gauge theories. Giving a meaning to this
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last concept in QFT on QST might thus be an approach to that basic question on the fate of Locality. But the problem we just mentioned is a main unsolved problem of the Algebraic Approach to Quantum Field Theory on the classical spacetimes as well: what is the signature of Gauge Theories with minimal interactions in terms of local algebras of observables alone? The status of Gauge Theories is mathematically crystal clear in the case of classical fields; not so much for Quantum Fields; but, anyway, it has a meaning only in terms of non observable fields (as the electromagnetic potentials and the charged Dirac Fields in Quantum Electrodynamics), and not in terms of local observables. This is an old question, posed long ago by Rudolf Haag, and mentioned in many discussions we had jointly, in particular with Dick as well, back in the early ’70ies. An answer to it might be the premise of a variant of that same answer which is valid on Quantum Spacetime, a variant where the seeds of Locality already present at the Planck scale might be hidden. We close with a comment on the concept of algebras of local observables, so central in the Algebraic approach to Quantum Field Theory. It is usually expressed by the map O → A(O) from (nice, bounded) regions in spacetime to von Neumann algebras generated by the observables which can be measured within the spacetime limitations expressed by those regions. But specifying a region is no longer meaningful in Quantum Spacetime. Observables localized near a point ought to be replaced by observables localized near a quantum point, which is specialized by a pure state ω on the C* algebra E of Quantum Spacetime. A region (classically described by its characteristic function, a selfadjoint idempotent of some algebra of Borel functions) ought to be replaced by a selfadjoint idempotent E in the Borel completion of E, and the observables localized at quantum points belonging to E (“points” described by optimally localized states ω such that ω(E) = 1) ought to generate the von Neumann algebra A(E); in a theory of an observable field φ, A(E) would include the operators associated to such pure states ω, as < ι ⊗ ω, φ(q) >, ω(E) = 1.
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The map E → A(E) would generate the quasilocal C* algebra A, and describe local algebras - but not for all such projections E. There should be a counterpart of the usual limitation to bounded open regions. This is a delicate point. A minimal choice would be the support projection of a state which is optimally localized at a point a. That is the spectral projection associated to the characteristic function of the point 2 for the operator Σµ ( qµ −aµ )2 multiplied with the spectral projection of the Qµν at a point of Σ1 . For that choice there is just one state ω such that ω(E) = 1, and the the local algebras might be rather small. But if E contains all translates of a non zero projection with translations in a neighborhood of the identity, the associated local algebra might be irreducible. This possibility, anticipated by Rudolf Haag, has been verified in the present and in related models for the free field (unpublished work with Klaus Fredenhagen, and [14]).
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References
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The map E → A(E) would be possibly covariant for global symmetry groups. In that case the map A ought to intertwine the actions of that group on the C* algebra A and on the Borel completion of E. Thus covariance would have a sharp and precise counterpart concerning global symmetry groups; but the fate of Covariance under general coordinate transformations (what would be their meaning down to the Planck scale? in Physical terms: what would the Einstein thought experiment of the free falling lift mean if the size of the lift becomes Planckian?) and of the Locality postulate, unfortunately, is not so clear.
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[1] Sergio Doplicher, Gerardo Morsella, Nicola Pinamonti: “Quantum Spacetime and the Universe at the Big Bang, vanishing interactions and fading degrees of freedom”; arXiv:1911.04756; [2] Sergio Doplicher, Gerardo Morsella, Nicola Pinamonti: “Perturbative Algebraic Quantum Field Theory on Quantum Spacetime: Adiabatic and Ultraviolet Convergence”; arXiv:1906.05855; [3] Sergio Doplicher, Gerardo Morsella, Nicola Pinamonti: “On Quantum Spacetime and the horizon problem”, J. Geom. Phys. 74 (2013), 196-210; [4] Sergio Doplicher, Klaus Fredenhagen, Gerardo Morsella, Nicola Pinamonti: “Pale Glares of Dark Matter in Quantum Spacetime”, Phys. Rev. D 95, 065009 (2017); [5] D. Bahns , S. Doplicher, G.Morsella, G. Piacitelli: “Quantum Spacetime and Algebraic Quantum Field Theory”, arXiv:1501.03298; in “Advances in Algebraic Quantum Field Theory”, R. Brunetti, C. Dappiaggi, K. Fredenhagen, J. Yngvason, Eds. ; Springer, 2015 [6] Sergio Doplicher, Klaus Fredenhagen, John E. Roberts: “The quantum structure of spacetime at the Planck scale and quantum fields”, Commun.Math.Phys. 172 (1995) 187-220; [7] Dorothea Bahns, Sergio Doplicher, Klaus Fredenhagen, Gherardo Piacitelli: “Quantum Geometry on Quantum Spacetime: Distance, Area and Volume Operators”, Commun. Math. Phys. 308, 567-589 (2011); [8] D. Bahns , S. Doplicher, K. Fredenhagen, G. Piacitelli: “Ultraviolet Finite Quantum Field Theory on Quantum Spacetime”, Commun.Math.Phys.237:221-241, 2003. [9] Luca Tomassini, Stefano Viaggiu: “Physically motivated uncertainty relations at the Planck length for an emergent non commutative spacetime”, Class. Quantum Grav. 28 (2011) 075001; [10] Luca Tomassini, Stefano Viaggiu: “Building non commutative spacetimes at the Planck length for Friedmann flat cosmologies”, arXiv:1308.2767; [11] Sergio Doplicher: “The Principle of Locality. Effectiveness, fate and challenges”, Journ. Math. Phys. 2010, 50th Anniversary issue;
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[12] E. G. Adelberger, B. R. Heckel and A. E. Nelson, Ann. Rev. Nucl. Part. Sci. 53, 77 (2003). arXiv:hep-ph/0307284; [13] C.D. Hoyle, D.J. Kapner, B.R. Heckel, E.G. Adelberger, J.H. Gundlach, U. Schmidt, H.E. Swanson: “Sub-millimeter Tests of the Gravitational Inverse-square Law”, Journal-ref: Phys.Rev.D70:042004,2004 arXiv:hep-ph/0405262; [14] Claudio Perini, Gabriele Nunzio Tornetta: “A scale-covariant quantum spacetime”, Rev. Math. Phys. 26, 1450006 (2014).
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ABSTRACT
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The Principles of Quantum Mechanics and of Classical General Relativity indicate that Spacetime in the small (Planck scale) ought to be described by a noncommutative C* Algebra, implementing spacetime uncertainty relations. A model C* algebra of Quantum Spacetime and its Quantum Geometry is described. Interacting Quantum Field Theory on such a background is discussed, with open problems and recent progress. Applications to cosmology suggest that the Planck scale ought to depend upon dynamics, and possible consequences in the large of the quantum structure in the small are outlined.
PII: DOI: Reference:
S0723-0869(20)30002-5 https://doi.org/10.1016/j.exmath.2020.01.002 EXMATH 25383
To appear in:
Expositiones Mathematicae
Please cite this article as: S. Doplicher, Quantum Spacetime, Quantum Geometry and Planck scales, Expositiones Mathematicae (2020), doi: https://doi.org/10.1016/j.exmath.2020.01.002. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. c 2020 Published by Elsevier GmbH. ⃝
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In memoriam Dick Kadison
Sergio Doplicher
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Quantum Spacetime, Quantum Geometry and Planck scales
Dipartimento di Matematica, Universit` a di Roma “La Sapienza”, Piazzale Aldo Moro, 5, I-00185 Roma, Italy, email [email protected].
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Introduction
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Noncommutativity is the main feature of Operator Algebras, a domain where Dick Kadison has been a giant. For these reason maybe it is not unappropriate to honor his memory talking of a new aspect of noncommutativity, which emerged in Physics in relatively recent times; after some heuristic motivation in this introduction, we outline some of its mathematical and physical implications in the next sections. It is well known that noncommutativity is the central feature of Quantum Mechanics; for systems with finitely many degrees of freedom, the Heisenberg uncertainty principle ∆q∆p & ~ relating the uncertainty in the measurement of a coordinate q of the position of a particle to that of its conjugate momentum p, are a basic physical principle, which is mathematically implemented by the Born - Heisenberg commutation relations pq − qp = −i~.
In Quantum Field Theory the basic observable quantities are the local observables: those which can be measured within the spacetime specifications given by a bounded open region (say a double cone) O in spacetime, generate a von Neumann algebra A(O), acting on the Hilbert space of the vacuum superselection sector, where the C* algebra generated for all O acts irreducibly. The basic principle of Locality requires that
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AB = BA
whenever
A ∈ A(O1 ), B ∈ A(O2 , ),
and O1 , O2 are spacelike separated, i.e. they cannot be connected by signals traveling not faster than light. This principle is consistent with what can be observed at all accessible scales; in Quantum Field Theory it holds at all scales, if we neglect the gravitational forces
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between elementary particles (which, of course, at today accessible distances, are weak beyond any imaginable level). If we do not, we must remember that, according to the Heisenberg uncertainty principle, localizing an event in a small region costs energy, and that, according to Einstein, energy generates a gravitational field. Thus the concurrence of Quantum Mechanics and of General Relativity leads us to the P rinciple of Gravitational Stability against localization of events [6]: The gravitational field generated by the concentration of energy required by the Heisenberg Uncertainty Principle to localize an event in spacetime should not be so strong to hide the event itself to any distant observer - distant compared to the Planck scale. To indicate how this principle limits the precision in localizing an event, recall that a spherically symmetric localization in space with accuracy a implies that an uncontrollable energy E of order 1/a has to be transferred (we use universal units where the Plank constant ~, the speed of light c, and the gravitational constant G are equal to 1). The associated Schwarzschild radius R, giving the size of a sphere around the origin out of which no signal can emerge, is R'E+U
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if U is the energy already present at the observed spot, in a background spherically symmetric quantum state. Hence we must have that a & R ' 1/a + U ;
so that if U is much smaller than 1
a & 1,
the Planck Length; that means, in CGS units,
a & λP ' 1.6 · 10−33 cm;
if U is much larger than 1,
a & U,
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and the “minimal distance” is dynamical, the Effective Planck Length, which might diverge. Which shows a possible path to solve the “Horizon Problem” (why the Cosmic Microwave Background has the same temperature in all directions, showing the reach of thermodynamical equilibrium even for regions which were never connected by non - superluminal signals?): divergent Effective Planck Length means instant long range (a-causal) correlations, allowing establishment of thermal equilibrium [3]. Such an explanation would not need the hypothesis of inflation.
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But at t = 0 all points are instantly connected to one another: a single point. The number of degrees of freedom is collapsing to zero? This point is discussed in [1]. Anyway, at least at distances of the order of the Planck length λP , causality, hence Locality, is expected to fail. The paradise, where an impressive part of the mathematical and conceptual structure of Quantum Field Theory derives essentially from Locality alone [11], is thus lost. One might hope that, at the Planck scale, a yet unknown but equally sharp Physical Principle replaces Locality, reducing to it at distances which are large compared with the Planck Length. We comment on this point at the end of this note. Let us now focus on the case where the local energy content U of the background state can be neglected, but we do not limit ourselves to the case of spherical symmetry. If we measure one or at most two space coordinates with great precision a, but the uncertainty L in another coordinates is large, the energy 1/a may spread over a region of size L, and generate a gravitational potential that vanishes everywhere as L → ∞ (provided a, as small as we like but non zero, remains constant). This indicates that the ∆q µ must satisfy Uncertainty Relations. Which should be then implemented by commutation relations. Giving rise to a Quantum Spacetime. But if we do not neglect the background energy U , there must be a dependence of the Uncertainty Relations, hence of the Commutators between coordinates, upon the background quantum state i.e. upon the metric tensor. Thus, while Classical General Relativity says that Geometry is Dynamics, a Quantum theory of Gravity is to be expected to say that also the Algebra of Spacetime is Dynamics.
Quantum Minkowski Space
The Principle of Gravitational Stability against localization of events implies the following Spacetime Uncertainty Relations (STUR):
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∆q0 ·
3 X j=1
X
∆qj & 1;
∆qj ∆qk & 1.
(1)
1≤j
first deduced using the linear approximation of the Einstein Equations [6]; adopting instead the Hoop Conjecture, further evidence was provided [9, 10], extending them to a curved background as well. The notion of Energy, problematic in General Relativity, plays a role: but in the special case of spherically symmetric experiments, with all spacetime uncertainties
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taking all the same value, the exact semiclassical Einstein Equations, without any reference to energy, imply a minimal value of the proper length of order of the effective Planck length [3]. We come back to this point further below. Returning to Minkowski, the STUR must be implemented by spacetime commutation relations [qµ , qν ] = iλ2P Qµν
(2)
imposing Quantum Conditions on the Qµν . The simplest solution: [q µ , Qν,λ ] = 0; µν
Qµν Q
(3)
= 0;
(4)
2
((1/2) [q0 , . . . , q3 ]) = I,
(5)
where, under Lorentz transformations, Qµν Qµν is a scalar, and
q0 .. [q0 , . . . , q3 ] ≡ det . q0
··· .. . ···
q3 .. . q3
≡ εµνλρ qµ qν qλ qρ =
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= −(1/2)Qµν (∗Q)µν
(6)
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is a pseudoscalar, hence we use the square in the Quantum Conditions. The so obtained “basic model of Quantum Spacetime” implements exactly the Spacetime Uncertainty Relations and is fully Poincar´e covariant. The classical Poincar´e group acts as symmetries; translations, in particular, act adding to each qµ a real multiple of the identity. The noncommutative C* algebra of Quantum Spacetime can be associated to the above relations. The procedure [6], which applies to more general cases, in short goes as follows. Assuming that the qλ , Qµν are selfadjoint operators and that the Qµν commute strongly with one another and with the qλ , the relations above can be seen as a bundle of Lie Algebra relations based on the joint spectrum Σ of the Qµν . Irreducible representations must be concentrated at points of Σ, providing representations of those Lie algebras; the regular representations must be integrable, hence described by representations of the C* group algebra of the unique simply connected Lie group associated to the corresponding Lie algebra. Thus the C* algebra of Quantum Spacetime is the C* algebra of a continuous field of group C* algebras based on the spectrum of a commutative C* algebra. In our case, that spectrum - the joint spectrum of the Qµν - is the manifold Σ of the real valued antisymmetric 2 - tensors fulfilling the same relations as the Qµν do: a homogeneous space of the proper orthochronous Lorentz group, identified with the coset space of SL(2, C) mod the subgroup of diagonal matrices. Each of
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those tensors, can be taken to its rest frame, where the “electric ”and “magnetic ”parts e, m are parallel unit vectors, by a boost. The inverse boost is specified by a third vector, orthogonal to those unit vectors; thus Σ can be viewed as the tangent bundle to two copies of the unit sphere in 3 space - its base Σ1 [5]. The irreducible representations at a point of Σ1 provide the Shroedinger operators p, q in 2 degrees of freedom. The fibers, with the condition that I is not an independent generator but is represented by I, coincide with the C* algebra of the Born - Heisenberg relations in 2 degrees of freedom - the algebra of all compact operators on a fixed infinite dimensional separable Hilbert space. The continuous field can be shown to be trivial. Thus the C* algebra E of Quantum Spacetime is identified with the tensor product of the continuous functions vanishing at infinity on Σ and the algebra of compact operators [6]. The mathematical generalization of points are pure states. The Optimally localized states ω are those minimizing the quantity Σµ (∆ω qµ )2 ;
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the minimum is 2, and is actually reached by states concentrated on Σ1 , at each point coinciding with the ground state of the harmonic oscillator . More precisely, these states are given by an optimal localization map composed with a probability measure on Σ1 . But to explore more thoroughly the Quantum Geometry of Quantum Spacetime we must consider independent events. Quantum mechanically n independent events ought to be described by the n − f old tensor product of E with itself; considering arbitrary values of n we are led to use the direct sum over all n. If A is the C* algebra with unit over C, obtained adding the unit to E, we will view the n-fold tensor power Λn (A) of A over C as an A-bimodule with the product in A, a(a1 ⊗ a2 ⊗ ... ⊗ an ) = (aa1 ) ⊗ a2 ⊗ ... ⊗ an ;
(a1 ⊗ a2 ⊗ ... ⊗ an )a = a1 ⊗ a2 ⊗ ... ⊗ (an a);
and the direct sum
Λ(A) =
∞ M
Λn (A)
n=0
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as the A-bimodule tensor algebra,
(a1 ⊗ a2 ⊗ ... ⊗ an )(b1 ⊗ b2 ⊗ ... ⊗ bm ) = a1 ⊗ a2 ⊗ ... ⊗ (an b1 ) ⊗ b2 ⊗ ... ⊗ bm .
This is the natural ambient for the universal differential calculus, where the differential is given by d(a0 ⊗ · · · ⊗ an ) =
n X
(−1)k a0 ⊗ · · · ⊗ ak−1 ⊗ I ⊗ ak ⊗ · · · ⊗ an .
k=0
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As usual d is a graded differential , i.e., if φ ∈ Λ(A), ψ ∈ Λn (A), we have
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d2 = 0;
d(φ · ψ) = (dφ) · ψ + (−1)n φ · dψ.
Note that A = Λ1 (A) ⊂ Λ(A), and the d-stable subalgebra Ω(A) of Λ(A) generated by A is the universal differential algebra with dI = 0. In other words, it is the subalgebra generated by A and da = I ⊗ a − a ⊗ I
as a varies in A. In the case of n independent events one is led to describe the spacetime coordinates of the j − th event by qj = I ⊗ ... ⊗ I ⊗ q ⊗ I... ⊗ I (q in the j - th place); in this way, the commutator between the different spacetime components of the qj would depend on j. A better choice is to require that it does not; this is achieved as follows. The centre Z of the multiplier algebra of E is the algebra of all bounded continuous complex functions on Σ; so that E, and hence A, is in an obvious way a Z − bimodule. We therefore can, and will, replace, in the definition of Λ(A), the C - tensor product, by the Z − bimodule − tensor product, so that
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dQ = 0.
As a consequence, the qj and the 2−1/2 (qj − qk ), j different from k, and the 2−1/2 dq, obey the same spacetime commutation relations, as does the normalized barycenter coordinates, n−1/2 (q1 + q2 + ... + qn ); and the latter commute with the difference coordinates. These facts allow us to define a quantum diagonal map from Λn (E) to E1 (the restriction to Σ1 of E), E (n) : E ⊗Z · · · ⊗Z E −→ E1
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which factors to that restriction map and a conditional expectation, which leaves the functions of the barycenter coordinates alone, and evaluates on functions of the difference variables the universal optimally localized map (which, when composed with a probability measure on Σ1 , would give the generic optimally localized state). Replacing the classical diagonal evaluation of a function of n arguments on Minkowski space by the quantum diagonal map allows us to define the Quantum W ick P roduct. But working in Ω(A) as a subspace of Λ(A) allows us to use two structures: - the tensor algebra structure described above, where both the A bimodule and the Z bimodule structures enter, essential for our reduced universal differential calculus; - the pre - C* algebra structure of Λ(A), which allows us to consider, for each element a of Λn (A), its modulus (a∗ a)1/2 , its spectrum, and so on.
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In particular we can study the geometric operators: separation between two independent events, area, 3 - volume, 4 - volume, given by dq;
dq ∧ dq;
dq ∧ dq ∧ dq;
dq ∧ dq ∧ dq ∧ dq, where, for instance, the latter is given by
V = dq ∧ dq ∧ dq ∧ dq = µνρσ dq µ dq ν dq ρ dq σ .
Each of these forms has a number of spacetime components: e.g. four for the first one (a vector), the last one is a pseudoscalar. Each component is shown to be a normal operator . Then we have the following Theorem [7]
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For each of these forms, the sum of the square moduli of all spacetime components is bounded below by a multiple of the identity of unit order of magnitude.
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Although that sum is (except for the 4 - volume!) not Lorentz invariant, the above mentioned bound holds in any Lorentz frame. In particular, - the four volume operator has pure point spectrum, at distance 51/2 − 2 from 0; - the Euclidean distance between two independent events has a lower bound of order one in Planck units. Thus, two distinct points can never merge to a point. However, of course, the state where the minimum is achieved will depend upon the reference frame where the requirement is formulated. (More details on the foregoing discussion may be found in [7] where the structure of length, area and volume operators on QST has been studied in full detail). Thus the existence of a minimal length is not at all in contradiction with the Lorentz covariance of the model. In the hope to use this formalism for Gauge Theory, a pairing was introduced on Λ(A) such that ha1 ⊗ a2 ⊗ .... ⊗ an , b1 ⊗ b2 ⊗ ... ⊗ bm i = δn,m T r(a1 b1 ...an bn )
where Tr is a faithful trace on A (here, any algebra with unit - and with a trace). Then, for any pair of differential forms φ, ψ on A,
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hφ, dψi = hδφ, ψi,
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where δ is the Hochschild boundary; which is then the Hodge dual of the exterior differential with respect to the above metric [7]. But unfortunately so far this formalism did not provide new insights on Gauge Theories on Quantum Spacetime. In the C* algebra E of Quantum Spacetime, we can define [6] the von Neumann functional calculus: for each f ∈ FL1 (R4 ) the function f (q) of the quantum coordinates qµ is given by Z µ f (q) ≡ fˇ(α)eiqµ α d4 α , and the integral over the whole space as well as that over 3 - space at q0 = t by
Z
Z
q0 =t
d4 qf (q) ≡ 3
f (q)d q ≡
Z
Z
f (x)d4 x = fˇ(0) = T rf (q), eik0 t fˇ(k0 , ~0)dk0 =
= lim T r(fm (q)∗ f (q)fm (q)), m
3
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where the trace is the ordinary trace at each point of the joint spectrum Σ of the commutators, i.e. a Z valued trace, and for suitable sequences of functions fm ∈ FL1 (R4 ). But on more general elements of our algebra both maps give Q - dependent results. These are important tools to define the interaction Hamiltonian to be used in the Gell’Mann Low formula for the S - Matrix.
Quantum Spacetime and Quantum Field Theory
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The geometry of Quantum Spacetime and the free field theories on it are fully Poincar´e covariant. One can introduce interactions in different ways, all preserving spacetime translation and space rotation covariance, all equivalent on ordinary Minkowski space, providing inequivalent approaches on QST; but all of them, sooner or later, meet problems with Lorentz covariance, apparently due to the nontrivial action of the Lorentz group on the centre of the multiplier algebra of Quantum Spacetime. On this points in our opinion a deeper understanding is needed. For instance, the interaction Hamiltonian on quantum spacetime Z HI (t) = λ d3 q : φ(q)n : q 0 =t
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would be Q - dependent; since there is no invariant probability measure or mean on Σ, we can do no better than integrating on Σ1 with the rotation invariant measure [6], and this breaks Lorentz invariance. Covariance is preserved by the Yang Feldmann equations, but missed again at the level of scattering theory. The Quantum Wick product selects a special frame from the start. The interaction Hamiltonian on the quantum spacetime is then given by Z HI (t) = λ d3 q : φ(q)n :Q q 0 =t
where
: φn (q) :Q = E (n) (: φ(q1 ) · · · φ(qn ) :)
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which does not depend on Q any longer, but brakes Lorentz invariance at an earlier stage. The last mentioned approach takes into account, in the very definition of Wick products, the fact that in our Quantum Spacetime n (larger or equal to two) distinct points can never merge to a point. But we can use the canonical quantum diagonal map which associates to functions of n independent points a function of a single point, evaluating a conditional expectation which on functions of the differences takes a numerical value, associated with the minimum of the Euclidean distance in a given Lorentz frame. The “Quantum Wick Product” obtained by this procedure leads to an interaction Hamiltonian on the quantum spacetime, given by the formula reported few line above, which is a constant operator–valued function of Σ1 (i.e. HI (t) is formally in the tensor product of C(Σ1 ) with the algebra of field operators). Thus one is lead to a unique prescription for the interaction Hamiltonian on quantum spacetime. When used in the Gell’Mann Low perturbative expansion for the S - Matrix, this gives the same result as the effective non local Hamiltonian determined by the kernel exp
n 1 X µ 2 (4) 1 X aj . a δ − 2 j,µ j n j=1
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The corresponding perturbative Gell’Mann Low formula is then free of ultraviolet divergences at each term of the perturbation expansion [8] . However, those terms have a meaning only after a sort of adiabatic cutoff: the coupling constant should be changed to a function of time, rapidly vanishing at infinity, say depending upon a cutoff time T. But the adiabatic limit T → ∞ is difficult problem. The use of the Quantum Wick product was dictated by the inconsistency with the quantum structure of spacetime of setting equal to zero the difference of coordinates of independent events. But there is another way to take care of the same problem: to replace, in the expression for the interaction Lagrangean density, the
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field operator φ(x) at a point x of the classical Minkowski space, with the evaluation of the field φ at a quantum point of QST, namely with < ι ⊗ ωx , φ(q) >
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where φ(q) is affiliated to F ⊗ E, F is the algebra of fields, ι is the identity map on F, and ωx the state of E optimally localized at x. Applied to a polynomial interaction, this procedure leads to another effective interaction Lagrangean on classical Minkowski space, whose integral over the whole spacetime agrees with that obtained using the Quantum Wick Product. But now the methods of Perturbative Algebraic Quantum Field Theory can be applied, and the perturbation expansion of the field operators is not only still ultraviolet finite, but also, in this case, converging term by term to an adiabatic limit when the time cutoff is removed [2]. This important fact allows us to consider the following problems. First, can one prove the convergence term by term to an adiabatic limit of the perturbation expansion of the S - matrix itself? If so, suppose we apply this construction to the renormalized Lagrangean of a theory which is renormalizable on the ordinary Minkowski space, with the counter terms defined by that ordinary theory, and with finite renormalization constants depending upon the Planck length λP . Those constants ought to be calculated for a generic value of λP by the requirement that the divergent terms of the key irreducible divergent Feynmann diagrams are cancelled. When λP → 0 , do we get back the ordinary renormalized expansion on Minkowski space? This would allow us to take the result of the described construction as a source of predictions to be compared with observations (cf [2, 1] for more details).
Quantum Spacetime and cosmology
The heuristic argument in the Introduction above suggest that the commutators between coordinates ought to depend on gµ,ν ; in that scenario the equations of motion sould include those commutation relations Rµ,ν − (1/2)Rgµ,ν = 8πTµ,ν (ψ);
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Fg (ψ) = 0;
µ
[q , q ν ] = iQµ,ν (g);
so that Algebra is Dynamics. And we can expect a dynamical minimal length. Which, in particular, would be divergent near singularities. This could solve Horizon Problem, without inflationary hypothesis. How solid are these heuristic arguments? We may consider as a model a massless scalar field with a semiclassical coupling with gravity; namely, we use the Quantum Wick product to define the Energy -
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Momentum Tensor TQµ,ν (q); we focus on the semiclassical Einstein Equations,with source ω ⊗ ηx (TQµ,ν (q)), where ω is a KMS state and ηx is a state on E optimally localized at x. If we assume full spherical symmetry the exact solution is available, and shows that the minimal proper length is at least λP [3]. Hence these simplifying ansaetze imply a solution describing spacetime without the horizon problem [3]. Near the Big Bang every pair of points were in causal contact, as indicated by the heuristic argument that the range of a-causal effects should diverge. For the Planck length λP is replaced by the effective Planck length λP /a(t), where a(t) is the coefficient in the FRW metric ds2 = −dt2 + a(t)2 (dx21 + dx22 + dx23 )
What happens at the singularity, t = 0? If we consider the Minkowski QST with varying effective Planck length, λeff → ∞,
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replacing λP by λeff → ∞ in the above formulas we get that, in that limit, the Quantum Diagonal Map vanishes: E (n) → 0. Interactions and the interacting fields themselves ought to vanish, a quantum point would be infinitely extended coinciding with the whole; a whole with zero degrees of freedom? This point and further consequences are discussed in [1]. But a fully convincing answer would require a dynamical picture of Quantum Spacetime, maybe in the scenario outlined at the beginning of this Section. Which might provide an answer to the question: are there observable signatures of QST? So far they seem too weak to be detected [4]. However, it has to be stressed that Quantum effects at Planck scale would depend upon an extrapolation of the Einstein Equations to that scale. But Newton’s law is experimentally checked only for distances not less than .01 centimeter! [12, 13], i.e. we are extrapolating 31 steps down in base 10 - logarithmic scale; while the size of the known universe is ”only” 28 steps up. Note that the observations with LHC in CERN on one side, and, on the other side, the nearly perfect agreement with Planck’s law of the spectral energy distribution of the Cosmic Microwave Background Radiation - the largest system in the universe - indicate the agreement with Quantum Mechanics of what can be observed in the range from 10−18 to 1028 centimeters. In the same range the Principle of Locality seems to hold unrestrictedly. But we saw it has to be expected that it will fail at the very short Planckian distances or near the big Bang; that is much below that lower limit, or about if not beyond that higher limit (for, due to its expansion, observing the known universe near its border would amount to observing it near the Big Bang). What would replace it? It might be helpful to wonder about the physical basis of that principle. Which seems to be the point - nature of field interactions, the great revolution of Maxwell’s viewpoint. That point - nature, in turn, is a consequence of the minimal form of interactions in gauge theories. Giving a meaning to this
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last concept in QFT on QST might thus be an approach to that basic question on the fate of Locality. But the problem we just mentioned is a main unsolved problem of the Algebraic Approach to Quantum Field Theory on the classical spacetimes as well: what is the signature of Gauge Theories with minimal interactions in terms of local algebras of observables alone? The status of Gauge Theories is mathematically crystal clear in the case of classical fields; not so much for Quantum Fields; but, anyway, it has a meaning only in terms of non observable fields (as the electromagnetic potentials and the charged Dirac Fields in Quantum Electrodynamics), and not in terms of local observables. This is an old question, posed long ago by Rudolf Haag, and mentioned in many discussions we had jointly, in particular with Dick as well, back in the early ’70ies. An answer to it might be the premise of a variant of that same answer which is valid on Quantum Spacetime, a variant where the seeds of Locality already present at the Planck scale might be hidden. We close with a comment on the concept of algebras of local observables, so central in the Algebraic approach to Quantum Field Theory. It is usually expressed by the map O → A(O) from (nice, bounded) regions in spacetime to von Neumann algebras generated by the observables which can be measured within the spacetime limitations expressed by those regions. But specifying a region is no longer meaningful in Quantum Spacetime. Observables localized near a point ought to be replaced by observables localized near a quantum point, which is specialized by a pure state ω on the C* algebra E of Quantum Spacetime. A region (classically described by its characteristic function, a selfadjoint idempotent of some algebra of Borel functions) ought to be replaced by a selfadjoint idempotent E in the Borel completion of E, and the observables localized at quantum points belonging to E (“points” described by optimally localized states ω such that ω(E) = 1) ought to generate the von Neumann algebra A(E); in a theory of an observable field φ, A(E) would include the operators associated to such pure states ω, as < ι ⊗ ω, φ(q) >, ω(E) = 1.
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The map E → A(E) would generate the quasilocal C* algebra A, and describe local algebras - but not for all such projections E. There should be a counterpart of the usual limitation to bounded open regions. This is a delicate point. A minimal choice would be the support projection of a state which is optimally localized at a point a. That is the spectral projection associated to the characteristic function of the point 2 for the operator Σµ ( qµ −aµ )2 multiplied with the spectral projection of the Qµν at a point of Σ1 . For that choice there is just one state ω such that ω(E) = 1, and the the local algebras might be rather small. But if E contains all translates of a non zero projection with translations in a neighborhood of the identity, the associated local algebra might be irreducible. This possibility, anticipated by Rudolf Haag, has been verified in the present and in related models for the free field (unpublished work with Klaus Fredenhagen, and [14]).
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References
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The map E → A(E) would be possibly covariant for global symmetry groups. In that case the map A ought to intertwine the actions of that group on the C* algebra A and on the Borel completion of E. Thus covariance would have a sharp and precise counterpart concerning global symmetry groups; but the fate of Covariance under general coordinate transformations (what would be their meaning down to the Planck scale? in Physical terms: what would the Einstein thought experiment of the free falling lift mean if the size of the lift becomes Planckian?) and of the Locality postulate, unfortunately, is not so clear.
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[1] Sergio Doplicher, Gerardo Morsella, Nicola Pinamonti: “Quantum Spacetime and the Universe at the Big Bang, vanishing interactions and fading degrees of freedom”; arXiv:1911.04756; [2] Sergio Doplicher, Gerardo Morsella, Nicola Pinamonti: “Perturbative Algebraic Quantum Field Theory on Quantum Spacetime: Adiabatic and Ultraviolet Convergence”; arXiv:1906.05855; [3] Sergio Doplicher, Gerardo Morsella, Nicola Pinamonti: “On Quantum Spacetime and the horizon problem”, J. Geom. Phys. 74 (2013), 196-210; [4] Sergio Doplicher, Klaus Fredenhagen, Gerardo Morsella, Nicola Pinamonti: “Pale Glares of Dark Matter in Quantum Spacetime”, Phys. Rev. D 95, 065009 (2017); [5] D. Bahns , S. Doplicher, G.Morsella, G. Piacitelli: “Quantum Spacetime and Algebraic Quantum Field Theory”, arXiv:1501.03298; in “Advances in Algebraic Quantum Field Theory”, R. Brunetti, C. Dappiaggi, K. Fredenhagen, J. Yngvason, Eds. ; Springer, 2015 [6] Sergio Doplicher, Klaus Fredenhagen, John E. Roberts: “The quantum structure of spacetime at the Planck scale and quantum fields”, Commun.Math.Phys. 172 (1995) 187-220; [7] Dorothea Bahns, Sergio Doplicher, Klaus Fredenhagen, Gherardo Piacitelli: “Quantum Geometry on Quantum Spacetime: Distance, Area and Volume Operators”, Commun. Math. Phys. 308, 567-589 (2011); [8] D. Bahns , S. Doplicher, K. Fredenhagen, G. Piacitelli: “Ultraviolet Finite Quantum Field Theory on Quantum Spacetime”, Commun.Math.Phys.237:221-241, 2003. [9] Luca Tomassini, Stefano Viaggiu: “Physically motivated uncertainty relations at the Planck length for an emergent non commutative spacetime”, Class. Quantum Grav. 28 (2011) 075001; [10] Luca Tomassini, Stefano Viaggiu: “Building non commutative spacetimes at the Planck length for Friedmann flat cosmologies”, arXiv:1308.2767; [11] Sergio Doplicher: “The Principle of Locality. Effectiveness, fate and challenges”, Journ. Math. Phys. 2010, 50th Anniversary issue;
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[12] E. G. Adelberger, B. R. Heckel and A. E. Nelson, Ann. Rev. Nucl. Part. Sci. 53, 77 (2003). arXiv:hep-ph/0307284; [13] C.D. Hoyle, D.J. Kapner, B.R. Heckel, E.G. Adelberger, J.H. Gundlach, U. Schmidt, H.E. Swanson: “Sub-millimeter Tests of the Gravitational Inverse-square Law”, Journal-ref: Phys.Rev.D70:042004,2004 arXiv:hep-ph/0405262; [14] Claudio Perini, Gabriele Nunzio Tornetta: “A scale-covariant quantum spacetime”, Rev. Math. Phys. 26, 1450006 (2014).
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ABSTRACT
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The Principles of Quantum Mechanics and of Classical General Relativity indicate that Spacetime in the small (Planck scale) ought to be described by a noncommutative C* Algebra, implementing spacetime uncertainty relations. A model C* algebra of Quantum Spacetime and its Quantum Geometry is described. Interacting Quantum Field Theory on such a background is discussed, with open problems and recent progress. Applications to cosmology suggest that the Planck scale ought to depend upon dynamics, and possible consequences in the large of the quantum structure in the small are outlined.